Hybrid symbolic‐numeric algorithms for computational convex analysis

Abstract

AbstractComputational convex analysis focuses on developing efficient tools to compute fundamental transforms arising in convex analysis. Symbolic computation tools have been developed, and have allowed more insight into the calculation of the Fenchel conjugate and related transforms. When such tools are not applicable e.g. when there is no closed form, fast transform algorithms perform numerical computation efficiently.However, computing the composition of several transforms is difficult to achieve with fast transform algorithms, which is the case for the recently introduced proximal average operator. We consider the class of piecewise linear‐quadratic functions which, being closed under the most relevant operations in convex analysis, allows the robust and efficient numerical computation of compositions of transforms like the proximal average. The algorithms presented are hybrid symbolic‐numeric: they first compute a piecewise linear‐quadratic approximation of the function, and then manipulate the approximation symbolically. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)